Remark 3.13 Let \((g,K)\)  be an almost-para-Hermitian structure, let \(P\)  and \(\tilde{P}\)  be the associated projectors
(REF ) to its eigenbundles, and let \(\alpha ^{A},\beta ^{A}\)  be the spinor fields representing \(K\)  (eq. (REF )).
Furthermore, let \(\nabla \)  be an arbitrary derivative operator, and \(X\in \Gamma (TM\otimes \mathbb {C})\) .
In the modern para-Hermitian approach to double field theory, see in particular
[1]} and [2]}, an especially important role is played by the operators \(\nabla _{\tilde{P}(X)}\)  and \(\nabla _{P(X)}\) 
(in arbitrary dimensions).
These operators are also crucial in the present work, and in our notation we have the equivalence:
\(& \tilde{\nabla }_{A^{\prime }}:=\alpha ^{A}\nabla _{AA^{\prime }} \quad \Leftrightarrow \quad \nabla _{\tilde{P}(X)}, \\& \nabla _{A^{\prime }}:=\beta ^{A}\nabla _{AA^{\prime }} \quad \Leftrightarrow \quad \nabla _{P(X)}.\) 
